348 XV NORMED ALGEBRAS AND SPECTRAL THEORY Let (xn)n^1 be a dense sequence in H. We define the Hn inductively, as follows. H! is the closure of the vector subspace of H generated by the U(s) • xl9 as s runs through A; we have xl e Hl9 because A has a unit element. If H!, ...,!!„_! have been constructed, it may be the case that H is equal to the (direct) sum L of the Hf, and the construction stops. If not, let U be the orthogonal supplement of L in H, and let p(n) be the smallest integer such that, if x'n is the orthogonal projection of xp^n) on L', the vector subspace H^ of L' generated by the U(s) - x'n is not equal to {0}: there exists such an index because the representation Uis nondegenerate, and we take Hn to be the closure of H^. Since x'neHn, the subspaces Hn satisfy the required conditions, by virtue of (6.4.2). (15.5.7) If A is a Banach algebra with involution, having a unit element, then every representation s*-+U(s) of A in a Hilbert space satisfies \\U(s)\\ <£ \\s\\ (and in particular C/is a continuous mapping of A into «Sf(H)). We have \\U(s)\\2 = \\U(s)*U(s)\\ = p(J7(j)*Z7(j)) in the star algebra j?(H) (15.4.14.1). Since U(sy*U(s) = U(s*s), it follows from (15.2.8Q) that In particular, we recover in this way (15.3.1(ii)). 6. POSITIVE LINEAR FORMS, POSITIVE HILBERT FORMS AND REPRESENTATIONS Let A be an algebra with involution (not necessarily normed, and not necessarily having a unit element). A linear form/: A -*• C on A is said to be positive if (15.6.1)' /(**jc)^0 for all jc e A. (15.6.2) Let fbe a positive linear form on an algebra A with involution. (i) The mapping (x, y)*-+ff(x9 y) =f(y*x) is a positive hermitian form on A x A: in other words, (15.6.2.1) /(**J>) =/(/**) for all x, y in A.ns through A, is dense in H. (In any case, this subspace is stable with